**Hint** $\rm\displaystyle\:\ \begin{align} f{\:'}\!\! &=\ \rm a\ f \\ \rm \:\ g'\!\! &=\ \rm a\ g \end{align} \iff \dfrac{f{\:'}}f = \dfrac{g'}g \iff \bigg(\!\!\dfrac{f}g\bigg)' =\ 0\ \iff W(f,g) = 0\:,\ \ W = $ Wronskian

This is a very special case of the *uniqueness* theorem for linear differential equations, esp. how the Wronskian serves to measure linear independence of solutions. See here for a proof of the less trivial second-order case (that generalizes to n'th order). See also the classical result below on Wronskians and linear dependence from one of my sci.math posts on May 12, 2003.

**Theorem** $\ \ $ Suppose $\rm\:f_1,\ldots,f_n\:$ are $\rm\:n-1\:$ times differentiable on interval $\rm\:I\subset \mathbb R\:$
and suppose they have Wronskian $\rm\: W(f_1,\ldots,f_n)\:$ vanishing at all points in $\rm\:I\:.\:$ Then $\rm\:f_1,\ldots,f_n\:$ are linearly dependent on some *subinterval* of $\rm\:I\:.$

**Proof** $\ $ We employ the following easily proved Wronskian identity:

$\rm\qquad\ W(g\ f_1,\ldots,\:g\ f_n)\ =\ g^n\ W(f_1,\ldots,f_n)\:.\ $ This immediately implies

$\rm\qquad\quad\ \ \ W(f_1,\ldots,\: f_n)\ =\ f_1^{\:n}\ W((f_2/f_1)',\ldots,\:(f_n/f_1)'\:)\quad $ if $\rm\:\ f_1 \ne 0 $

Proceed by induction on $\rm\:n\:.\:$ The Theorem is clearly true if $\rm\:n = 1\:.\:$ Suppose that $\rm\: n > 1\:$ and $\rm\:W(f_1,\ldots,f_n) = 0\:$ for all $\rm\:x\in I.\:$
If $\rm\:f_1 = 0\:$ throughout $\rm\:I\:$ then $\rm\: f_1,\ldots,f_n\:$ are dependent on $\rm\:I.\:$ Else $\rm\:f_1\:$ is nonzero at some point of $\rm\:I\:$ so also throughout some subinterval $\rm\:J \subset I\:,\:$ since $\rm\:f_1\:$ is continuous (being differentiable by hypothesis). By above $\rm\:W((f_2/f_1)',\ldots,(f_n/f_1)'\:)\: =\: 0\:$ throughout $\rm\:J,\:$ so by induction there exists a subinterval $\rm\:K \subset J\:$
where the arguments of the Wronskian are linearly dependent, i.e.

on $\rm\ K:\quad\ \ \ c_2\ (f_2/f_1)' +\:\cdots\:+ c_n\ (f_n/f_1)'\: =\ 0,\ \ $ all $\rm\:c_i'\:=\ 0\:,\ $ some $\rm\:c_j\ne 0 $

$\rm\qquad\qquad\: \Rightarrow\:\ \ ((c_2\ f_2 +\:\cdots\: + c_n\ f_n)/f_1)'\: =\ 0\ \ $ via $({\phantom m})'\:$ linear

$\rm\qquad\qquad\: \Rightarrow\quad\ \ c_2\ f_2 +\:\cdots\: + c_n\ f_n\ =\ c_1 f_1\ \ $ for some $\rm\:c_1,\ c_1'\: =\: 0 $

Therefore $\rm\ f_1,\ldots,f_n\:$ are linearly dependent on $\rm\:K \subset I\:.\qquad$ **QED**

This theorem has as immediate corollaries the well-known results
that the vanishing of the Wronskian on an interval $\rm\: I\:$ is
a necessary and sufficient condition for linear dependence of

$\rm\quad (1)\ $ functions analytic on $\rm\: I\:$

$\rm\quad (2)\ $ functions satisfying a monic homogeneous linear differential
equation

$\rm\quad\phantom{(2)}\ $ whose coefficients are continuous throughout $\rm\: I\:.\:$